Question

The position of a particle is given by $ r = 3.0t\hat i + 2.0{t^2}\hat j + 5.0\hat k $ where $ t $ is in seconds and the coefficients have their proper units for $ r $ to be in metres. Find $ v\left( t \right) $ and $ a(t) $ of the particle.

Answer 1

Hint : The velocity of a particle can be defined as the rate of change of position with time. Acceleration is the rate of change of velocity with time. So we need to differentiate the position with respect to time to find the velocity and then again differentiate the velocity with respect to time to find the acceleration.

Formula used: In this solution we will be using the following formula;
 $ v = \dfrac{{dr}}{{dt}} $ , where $ v $ is instantaneous, $ r $ is the position, and $ t $ is time.
 $ \dfrac{{dr}}{{dt}} $ signifies a differential rate of change of position with time.
 $ a = \dfrac{{dv}}{{dt}} $ where $ a $ is the acceleration, $ v $ is the velocity, and $ t $ is time.

Complete step by step answer
The position of a particle has been given as a particular function of time, and we are expected to find the velocity and acceleration as a function of time. This signifies that we are expected to find the instantaneous values of these quantities.
The instantaneous velocity is given by
 $ v = \dfrac{{dr}}{{dt}} $ where , $ r $ is the position, and $ t $ is time. $ \dfrac{{dr}}{{dt}} $ signifies a differential rate of change of position with time.
Hence, to calculate, we replace $ r $ in the equation with the expression given in question
 $ v = \dfrac{d}{{dt}}\left( {3.0t\hat i + 2.0{t^2}\hat j + 5.0\hat k} \right) $
Differentiating, we have
 $ v(t) = \left( {3.0\hat i + 4.0t\hat j} \right)m/s $ which is the velocity as a function of time.
Similarly, to calculate the instantaneous acceleration, we know that
 $ a = \dfrac{{dv}}{{dt}} $ where , $ v $ is the velocity, and $ t $ is time
Hence, by replacing the function of velocity that we have gotten from above we have,
 $ a = \dfrac{d}{{dt}}\left( {3.0\hat i + 4.0t\hat j} \right) $
By differentiating
 $ a = 4.0\hat jm/{s^2} $ .

Note
For clarity, the function $ v(t) $ and $ a(t) $ are called instantaneous velocity and acceleration because we can easily find the velocity and acceleration at any particular point (hence, the word instantaneous). For example, if we’re to find the velocity say at exactly the tenth second, we substitute 10 for $ t $ in the equation $ v(t) = 3.0\hat i + 4.0t\hat j $ as in:
 $ v\left( {10} \right) = 3.0\hat i + 4.0\left( {10} \right)\hat j $
 $ \Rightarrow v\left( {10} \right) = \left( {3.0\hat i + 40\hat j} \right)m/s $
But we can observe that the acceleration is not a function of time or can also be said to be a constant function of time. Hence, the acceleration of this body is constant (does not change with time).